We show that various aspects of k-automatic sequences -such as having an unbordered factor of length n -are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems.

a b s t r a c tWe revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given k-automatic sequence is ultimately periodic. We prove that it is decidable whether a given k-automatic sequence is overlap-free (or squarefree, or cubefree, etc.). We prove that the lexicographically least sequence in the orbit closure of a k-automatic sequence is k-automatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope α, have automatic continued fraction expansions if α does.

Abstract. We prove Dejean's conjecture. Specifically, we show that Dejean's conjecture holds for the last remaining open values of n, namely 15 ≤ n ≤ 26.

We show that Dejean's conjecture holds for n ≥ 27. This brings the final resolution of the conjecture by the approach of Moulin Ollagnier within range of the computationally feasible.

We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with |y| ≥ 4; our construction is somewhat simpler than the original construction of Dekking. Second, we construct an infinite binary word avoiding all squares except 0 2 , 1 2 , and (01) 2 ; our construction is somewhat simpler than the original construction of Fraenkel and Simpson. In both cases, we also show how to modify our construction to obtain exponentially many words of length n with the given avoidance properties. Finally, we answer an open question of Prodinger and Urbanek from 1979 by demonstrating the existence of two infinite binary words, each avoiding arbitrarily large squares, such that their perfect shuffle has arbitrarily large squares.

We give an O(n + t) time algorithm to determine whether an NFA with n states and t transitions accepts a language of polynomial or exponential growth. Given an NFA accepting a language of polynomial growth, we can also determine the order of polynomial growth in O(n+t) time. We also give polynomial time algorithms to solve these problems for context-free grammars. Time 599 reach a final state. For each state q ∈ Q, we define a new NFA M q = (Q, Σ, δ, q, {q}) and write L q = L(M q ).Following Ginsburg and Spanier, we say that a language L ⊆ Σ * is commutative if there exists u ∈ Σ * such that L ⊆ u * . The following lemma is essentially a special case of a stronger result for context-free languages (compare Theorem 5.5.1 of [7], or in the case of languages specified by DFA's, Lemmas 2 and 3 of [26]).Lemma 2. The language L(M ) has polynomial growth if and only if for every q ∈ Q, L q is commutative.We now are ready to prove Theorem 1.Proof. Let n denote the number of states of M . The idea is as follows. For every q ∈ Q, if L q is commutative, then there exists u ∈ Σ * such that L q ⊆ u * . For any w ∈ L q , we thus have w ∈ u * . If z is the primitive root of w, then z is also the primitive root of u. If L q ⊆ z * , then L q is commutative. On the other hand, if L q ⊆ z * , then L q contains two words with different primitive roots, and is thus not commutative. This argument leads to the following algorithm.Let q ∈ Q be a state of M . We wish to check whether L q is commutative. Any accepting computation of M q can only visit states in the strongly connected component of M containing q. We therefore assume that M is indeed strongly connected (if it is not, we run the algorithm on each strongly connected component separately; they can be determined in O(n + t) time (Section 22.5 of [4])).We first construct the NFA M q accepting L q . This takes O(n + t) time. Then we find a word w ∈ L(M q ), where |w| < n. If L(M q ) is non-empty, such a w exists and can be found in O(n+t) time. Next we find the primitive root of w, i.e., the shortest word z such that w = z k for some k ≥ 1. This can be done in O(n) time using any linear time pattern matching algorithm. To find the primitive root of w = w 1 · · · w , find the first occurrence of w in w 2 · · · w w 1 · · · w −1 . If the first occurrence begins at position i, then z = w 1 · · · w i is the primitive root of w.For i = 0, 1, . . . , |z| − 1, let A i be the set of all q ∈ Q such that there is a path from q to q labeled by a word from z * z 1 z 2 · · · z i . Observe that if some q belongs to A i and A j where i < j then we can find two different paths from q to q: z a z 1 · · · z i s and z b z 1 · · · z j s, where a and b are non-negative integers and s is the label of some path from q to q. For L q to be commutative, both these words must be powers of z, which is impossible: their lengths are different modulo |z|. Thus, the A i 's must be disjoint.We determine the A i 's as follows. To begin, we know q ∈ A 0 . For any i, if q ∈ A i , then we know q ∈ A (i+1) mod |z| for...

We show that the abelian complexity function of the ordinary paperfolding word is a 2-regular sequence. arXiv:1208.2856v1 [math.CO]

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